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Register a new userCollective mode reductions for populations of coupled noisy oscillators
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We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz, and (iii) a two-cumulant truncation of the circular cumulant representation of the original systems dynamics. For the latter we suggest a closure, which makes the truncation, for small noise, a rigorous first-order correction to the Ott-Antonsen ansatz, and simultaneously is a generalization of the Gaussian ansatz. The Kuramoto model with intrinsic noise, and the population of identical noisy active rotators in excitable states with the Kuramoto-type coupling, are considered as examples to test validity of these approximations. For all considered cases, the Gaussian ansatz is found to be more accurate than the Ott-Antonsen one for high-synchrony states only. The two-cumulant approximation is always superior to both other approximations.
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In this paper we present a systematic, data-driven approach to discovering bespoke coarse variables based on manifold learning algorithms. We illustrate this methodology with the classic Kuramoto phase oscillator model, and demonstrate how our manifold learning technique can successfully identify a coarse variable that is one-to-one with the established Kuramoto order parameter. We then introduce an extension of our coarse-graining methodology which enables us to learn evolution equations for the discovered coarse variables via an artificial neural network architecture templated on numerical time integrators (initial value solvers). This approach allows us to learn accurate approximations of time derivatives of state variables from sparse flow data, and hence discover useful approximate differential equation descriptions of their dynamic behavior. We demonstrate this capability by learning ODEs that agree with the known analytical expression for the Kuramoto order parameter dynamics at the continuum limit. We then show how this approach can also be used to learn the dynamics of coarse variables discovered through our manifold learning methodology. In both of these examples, we compare the results of our neural network based method to typical finite differences complemented with geometric harmonics. Finally, we present a series of computational examples illustrating how a variation of our manifold learning methodology can be used to discover sets of effective parameters, reduced parameter combinations, for multi-parameter models with complex coupling. We conclude with a discussion of possible extensions of this approach, including the possibility of obtaining data-driven effective partial differential equations for coarse-grained neuronal network behavior.
Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.
Dynamic-mode decomposition (DMD) is a versatile framework for model-free analysis of time series that are generated by dynamical systems. We develop a DMD-based algorithm to investigate the formation of functional communities in networks of coupled, heterogeneous Kuramoto oscillators. In these functional communities, the oscillators in the network have similar dynamics. We consider two common random-graph models (Watts--Strogatz networks and Barabasi--Albert networks) with different amounts of heterogeneities among the oscillators. In our computations, we find that membership in a community reflects the extent to which there is establishment and sustainment of locking between oscillators. We construct forest graphs that illustrate the complex ways in which the heterogeneous oscillators associate and disassociate with each other.
Coupled oscillators are shown to experience amplitude death for a much larger set of parameter values when they are connected with time delays distributed over an interval rather than concentrated at a point. Distributed delays enlarge and merge death islands in the parameter space. Furthermore, when the variance of the distribution is larger than a threshold the death region becomes unbounded and amplitude death can occur for any average value of delay. These phenomena are observed even with a small spread of delays, for different distribution functions, and an arbitrary number of oscillators.
A novel possibility of self-organized behaviour of stochastically driven oscillators is presented. It is shown that synchronization by Levy stable processes is significantly more efficient than that by oscillators with Gaussian statistics. The impact of outlier events from the tail of the distribution function was examined by artificially introducing a few additional oscillators with very strong coupling strengths and it is found that remarkably even one such rare and extreme event may govern the long term behaviour of the coupled system. In addition to the multiplicative noise component, we have investigated the impact of an external additive Levy distributed noise component on the synchronisation properties of the oscillators.
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